The past stability of Kasner singularities for the $(3+1)$-dimensional Einstein vacuum spacetime under polarized $U(1)$-symmetry

TL;DR

This paper provides a new proof of the past stability of Kasner singularities in 3+1 dimensional Einstein vacuum spacetimes with polarized U(1) symmetry, using a novel orthonormal-frame decomposition and Fuchsian techniques.

Contribution

It introduces a new method based on a 2+1 orthonormal-frame decomposition and symmetrization to analyze stability, extending previous results with a different approach.

Findings

Perturbed solutions are asymptotically Kasner near singularity

Solutions are geodesically incomplete and crushing at the Big Bang

Global existence and precise asymptotics are established up to singularities

Abstract

In this paper, we give a new proof to a past stability result established in Fournodavlos-Rodnianski-Speck (arXiv:2012.05888), for Kasner solutions of the $(3+1)$-dimensional Einstein vacuum equations under polarized $U(1)$-symmetry. Our method, inspired by Beyer-Oliynyk-Olvera-Santamar{\'\i}a-Zheng (arXiv:1907.04071, arXiv:2502.09210), relies on a newly developed $(2+1)$ orthonormal-frame decomposition and a careful symmetrization argument, after which the Fuchsian techniques can be applied. We show that the perturbed solutions are asymptotically pointwise Kasner, geodesically incomplete and crushing at the Big Bang singularity. They are achieved by reducing the $(3+1)$ Einstein vacuum equations to a Fuchsian system coupled with several constraint equations, with the symmetry assumption playing an important role in the reduction. Using Fuchsian theory together with finite speed of constraints propagation, we obtain global existence and precise asymptotics of the solutions up to the singularities.